I'd like to upload and thenunzip my zipped web site on my web host service which manage SSH connections too using Filezilla. I wonder if is it possible to use only that application to do both operations please. Thank you.
I checked 02 and 13 of the SFTP-related IETF drafts linked at http://wiki.filezilla-project.org/SFTP_specifications. There is no indication that it is possible to execute an arbitrary command via SFTP. It does not support compressed files by any means of knowing that the file is compressed either.
So the answer is no, that's not possible.
However, you can create your own "uncompress mechanism" if you want. If you have SSH access to the box, you can install